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Solving Systems of Linear Equations

A system of linear equations is formed when two or more linear equations are working together and they have same set of solutions. A same set of solution satisfies all equations in the given system.

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Solving Systems of Linear Equations

For solving system of linear equations we have various methods:
1)    Graphing equations
2)    Substitution method
3)    Elimination method
Let us solve a following system using all the above mentioned methods.

Consider a system: x + y = 10 and x – y = 4.

While solving this system by graphing method we just have to draw a graph of these two lines and the point of intersection of these two lines forms the solution of this system. Co-ordinates of the point of intersection gives the value of x and y. X co-ordinate represents value of x and Y coordinates represents the value of y. For this problem we get a point of intersection as (7, 3) that means x = 7 and y = 3.

We get the same solution for any other method.

Let us see how it works if we use substitution method.

Given system: x + y = 10 and x – y = 4
In the substitution method we convert any one equation in terms of a variable x or y i.e. we isolate x or y on single side.
Let us isolate x in first equation x + y = 10.
x = 10 – y
Now we substitute this expression for x in second equation x – y = 4.
(10 – y) – y = 4   implies that   10 – 2y = 4
Isolate y on left side.
-2y = 4 – 10 gives -2y = -6.
Dividing both sides by -2 gives y = 3.
Now we substitute 3 instead of y in any one equation and then solve for y.
x + 3 = 10    gives    x = 10 – 3 = 7.
Hence we get the same solutions as we get in graphing method, x = 7 and y = 3.

Now let us see what happens when we use elimination method.

In the elimination method we remove or eliminate any one variable. For this we have to first check whether the coefficient of a variable which is to be eliminated is same or not. It must be same in order to eliminate that variable. We can perform addition or subtraction for elimination whichever is appropriate. If we have equal and opposite variables then we do addition but if coefficients are equal with same signs then we do the subtraction.

Solved Example

Question: Solve system of equations by substitution
2x + 3y = 28   ....(1)
x - 2y = 7  ....(2)
Solution:

We have x - 2y = 7
x = 7 + 2y
Substitute value of x in equation 1
2 (7 + 2y) + 3y = 28
14 + 4y + 3y = 28
14 + 7y = 28
14 + 7y - 14 = 28 - 14
7y = 14
Hence y = 2
Now we can plug the value of y back in the equation for x
x = 7 + 2 (2)
= 7 + 4 = 11
x = 11, y = 2

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